every submetrizable space has a regular G_delta diagonal

Hello;

I proved the statement that every submetrizable space has a regular -diagonal. Kindly I want you to check if there is some mistake.

First a topological space has a regular -diagonal if the diagonal of can be expressed as a countable intersection of closure of open sets containing the diagonal, i.e. Δ . A space is called a submetrizable space if there is a continuous injective function from into a metric space.
I wrote the proof in the file attached. Kindly open the file to check the proof and give me your comments. Every guidance or comment is highly appreciated.

Re: every submetrizable space has a regular G_delta diagonal

Originally Posted by student2011

Hello;

I proved the statement that every submetrizable space has a regular -diagonal. Kindly I want you to check if there is some mistake.

First a topological space has a regular -diagonal if the diagonal of can be expressed as a countable intersection of closure of open sets containing the diagonal, i.e. Δ . A space is called a submetrizable space if there is a continuous injective function from into a metric space.
I wrote the proof in the file attached. Kindly open the file to check the proof and give me your comments. Every guidance or comment is highly appreciated.

Thaaaaaaaaank you in advance

Hey man, I'm going to level with you. In general people on a forum like this aren't going to take the time to fully read all this stuff and check it. This is not because we are lazy or rude, it's just that you are asking very technical, specific questions and for everyone here we'd have to google half the terms you are using, and it would be too tough. You're best bet is to ask these kind of questions on sites which are frequented by people who do research in fields that are tangent to your questions--they might automatically know the asnwer, or at least know enough of the terminology so that they are at least willing to try and answer your question. I would suggest math.stackexchange.com.